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| Mirrors > Home > MPE Home > Th. List > xmstopn | Structured version Visualization version GIF version | ||
| Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) | 
| isms.x | ⊢ 𝑋 = (Base‘𝐾) | 
| isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | 
| Ref | Expression | 
|---|---|
| xmstopn | ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isms.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
| 2 | isms.x | . . 3 ⊢ 𝑋 = (Base‘𝐾) | |
| 3 | isms.d | . . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
| 4 | 1, 2, 3 | isxms 24457 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) | 
| 5 | 4 | simprbi 496 | 1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 × cxp 5683 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 distcds 17306 TopOpenctopn 17466 MetOpencmopn 21354 TopSpctps 22938 ∞MetSpcxms 24327 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-xms 24330 | 
| This theorem is referenced by: imasf1oxms 24502 ressxms 24538 prdsxmslem2 24542 tmsxpsmopn 24550 xmsusp 24582 cmetcusp1 25387 minveclem4a 25464 minveclem4 25466 qqhcn 33992 rrhcn 33998 rrexthaus 34008 dya2icoseg2 34280 sitmcl 34353 | 
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